Chapter 7-8 (and 2.2) Review Worksheet 1. Which of these is the best candidate for a Binomial model?

a. the number of people we survey until we find someone who has taken Statistics b. the number of people we survey until we find two people who have taken Statistics c. the number of people in a class of 25 who have taken Statistics d. the number of aces in a five-card Poker hand e. the number of sodas students drink per day

2. Which of these is the best candidate for a Binomial model? a. the number of black cards in a 10-card hand b. the colors of the cars in Wegman’s parking lot c. the number of hits a baseball player gets in 6 times at bat d. the number of cards drawn from a deck until we find all four aces e. the number of people we survey until we find someone who owns an iPod

3. A far coin has come up “heads” 10 times in a row. The probability that the coin will come up heads on the next flip is

a. less than 50%, since “tails” is due to come up. b. 50% c. greater than 50%, since it appears that we are in a streak of heads. d. It cannot be determined.

4. Let X and Y be discrete random variables and let a and b be constants. Which of the following is false?

a. mean (X + Y) = mean (X) + mean (Y) b. mean (X – Y) = mean (X) – mean (Y) c. mean (aX) = (a)(mean (X)) d. mean (a + bX) = a + b mean X

e. If X and Y are independent, then standard deviation (X – Y) = 2 2X Yσ σ−

5. A randomly chosen subject arrives for a study of exercise and fitness. Consider these statements.

I. After 10 minutes on an exercise bicycle, you ask the subject to rate his or her effort on the Rate of Perceived Exertion (RPE) scale. RPE ranges in whole-number steps from 6 (no exertion at all) to 20 (maximum exertion).

II. You measure VO2, the maximum volume of oxygen consumed per minute during exercise. VO2 is generally between 2.5 liters per minute and 6 liters per minute.

III. You measure the maximum heart rate (beats per minute). The statements that describe a discrete random variable are a. None b. I. c. II. d. I, III. e. I, II, III. 6. Government statistics tell us that 2 out of every 3 American adults are overweight. Let X = number of Americans that

are overweight. How large would an SRS of American adults need to be in order for it to be safe to assume that the sampling distribution of X is approximately Normal? a. 3 b. 9 c. 15 d. 18 e. 30

7. The lifetime of a 2-volt non-rechargeable battery in constant use has a normal distribution with a mean of 516 hours

and a standard deviation of 20 hours. The proportion of batteries with lifetimes exceeding 520 hours is approximately a. 0.2000. b. 0.5793. c. 0.4207

8. The lifetime of a 2-volt non-rechargeable battery in constant use has a normal distribution with a mean of 516 hours

and a standard deviation of 20 hours. 90% of all batteries have a lifetime shorter than a. 517.28 hours. b. 536.00 hours. c. 541.50 hours

Answers: 1.C 2.C 3.B 4.E 5.D 6.E 7.C 8.C

9. If a distribution is relatively symmetric and mound-shaped, order from least to greatest, the following positions: a z-score of – 1, Q 1, a value in the 60th percentile

10. A company’s manufacturing process uses 500 gallons of water at a time. A “scrubbing” machine then removes most

of a chemical pollutant before pumping the water into a nearby lake. Legally the treated water should contain no more than 80 parts per million of the chemical, but the machine isn’t perfect and it is costly to operate. Since there’s a fine if the discharged water exceeds the legal maximum, the company sets the machine to attain an average of 75 ppm for the batches of water treated. They believe the machine’s output can be described by a Normal model with standard deviation 4.2 ppm.

a. What percent of the batches of water discharged exceed the 80 ppm standard? b. The company’s lawyers insist that they did not have more than 2% of the water over the limit. To what mean

value should the company set the scrubbing machine? Assume the standard deviation does not change. 11. You play two games against the same opponent. The probability that you win the first game is 0.4. If you win the

first game, the probability you also win the second is 0.2. If you lose the first game, the probability that you win the second is 0.3 a. Let X be the number of games you win. Find the probability distribution for X. b. What are the expected value and standard deviation of X?

12. A certain tennis player makes a successful first serve 70% of the time. Assume that each serve is independent of the

others. Suppose the tennis player serves 80 times in a match. a. What is the mean and standard deviation of the number of good first serves expected? b. Verify that you can use a Normal model to approximate the distribution of the number of good first serves. c. What is the probability that she makes at least 65 first serves? Use both a Normal approximation and a Binomial

calculation. 13. A local TV retailer always offers its customers an extended warranty to cover the costs of any repairs needed in the

next 3 years. Let x = the cost to repair a randomly selected TV under warranty. x 0 50 100 150 200 250 300

P(x) 0.71 0.13 0.08 0.04 0.02 0.01 0.01

a. Find and interpret the value of P(x = 0). b. Based on past history, the company has estimated the probability distribution of x. How much should the company

charge for the extended warranty? Explain your reasoning. c. What is the standard deviation of x? Interpret this value. d. If the retailer sold 5 TV’s with extended warranties, what are the mean and standard deviation of the total cost of

repairs for these 5 TV’s? 14. The owner of a small convenience store is trying to decide whether to discontinue selling magazines. He suspects that

only 5% of the customers buy a magazine and thinks that he might be able to use the display space to sell something more profitable. Before making a final decision, he decides that for one day he’ll keep track of the number of customer and whether or not they buy a magazine. a. Assuming that the owner is correct in thinking that 5% of the customers purchase magazines, how many customers

should he expect before someone buys a magazine? b. What is the probability that he does not sell a magazine until the 8th customer? c. What is the probability that exactly 2 of the first 10 customers buy magazines? d. What is the probability that at least 5 of his first 50 customers buy magazines? e. He had 280 customers that day. Assuming this day was typical for his store, what would be the mean and standard

deviation of the number of customers who buy magazines each day? 15. In an experiment comparing two weight loss plans, 100 subjects were randomly assigned to two groups, A and B.

The mean weight loss in group A was 10 pounds with a standard deviation of 8 pounds and the mean weight loss in group B was 7 pounds with a standard deviation of 11 pounds. The distributions of weight loss are approximately normal.

a. What is the probability that a randomly selected person from group A lost weight? b. What is the probability that a randomly selected person from group B gained weight?

c. If you were to randomly select 3 people from group A, what is the probability that they lost a total of 25 pounds or more?

d. If you were to randomly select one person from each group, what is the probability that the person from group B lost more weight?

e. What is the probability that a randomly selected person from group A lost at least 5 more pounds than a randomly selected person from group B?

f. In parts c-e, what assumption did you have to make? Is it reasonable?

• Additional problems to study: 7.55-58; 8.22, 24, 63, 65 • Know how to do all homework problems that were assigned and understand the questions from the quiz • Look through notes – understand not only how to calculate values, but how to interpret them as well.

Chapter 7-8 Review Worksheet ANSWERS 1. C 2. C 3. B 4. E 5. D 6. E 7. C 8. C 9. z score of 1, Q 1, 60% 10. a). 11.7%, b). 71.37 ppm 11. a. X = games won 0 1 2 P(X) 0.42 0.50 0.08

b. 0.66, 0.62 12. a. 56, 4.10 b. We can use a Normal approximation because np = 56 ≥ 10 and n(1-p) = 24 ≥ 10

c. Normal = 0.014; Binomial = 0.016 13. a. P(X = 0) = 0.71 The probability that a randomly selected TV costs $0 to repair under warranty is 0.71

b. The mean of X is $30. Thus, in the long run, the company should expect to pay an average of $30 for each TV under warranty. Because of this, the company should charge $30 or more for the extended warranty in order to not lose money in the long run.

c. $58.75 On average, the cost to repair a TV varies from the mean by about $58.75. d. mean = $150, S.D. = $131.35

14. a. 20 b. 0.035 c. 0.075 d. 0.104 e. 14, 3.65 15. a. 0.8944 b. 0.2623 c. 0.6409 P(X1 + X2 + X3 >= 25) d. 0.4127 P(XB > XA ) e. 0.4415 P(XA - XB >= 5)

f. We had to assume that the two variables (weight loss from group A and weight loss from group B) were independent. This is reasonable since the subjects in the experiment were randomly assigned.

Additional problems to study: 7.55-58; 8.22, 24, 48, 49, 53, 63, 65 7.56 – The insurance company is relying on the law of large numbers. Even though the company will lose a large amount of money on a small number of policyholders who die, it will gain a small amount from many thousands of 21-year old men. In the long run, the insurance company can expect to make $303.35 per insurance policy. 7.58a. mean = $303.35, S.D. = $6864.29 b. mean = $303.35, S.D. = $4853.78 8.22 Let X = number of home runs. Then X ~ B(509, 0.116) mean of X = 59.044 Using Normal approximation, X ~ N(59.044, 7.2246) and P(X >= 70) = 0.0643 8.24a. mean = 180, S.D. = 12.5857 b. Using Normal approximation, X ~ N(180, 12.5857) and P(165 < X < 195) = 0.766